Geometrical optics

Reflection and refraction

Reflection is the process where light bounces off a surface. In a mirror, light is almost completely reflected, with the angle of incidence equaling the angle of reflection. When light passes from one medium to another, part of it is reflected while the rest is transmitted.
Refraction occurs when light changes direction as it moves between media with different refractive indices ($n$). This bending is governed by Snell’s law, expressed as $n_{1}sin{\theta }_1$ = $n_{2}sin{\theta }_2$, where $n_1$ and $n_2$ represent the refractive indices of the respective media, and ${\theta }_1$ and ${\theta }_2$ are the angles measured from the normal.

Dispersion

Dispersion occurs because the refractive index of a medium varies with wavelength, causing different colors of light to bend by different amounts. As white light enters a prism, blue light (with a shorter wavelength) refracts more than red light (with a longer wavelength), resulting in the separation of colors into a spectrum.

Conditions for total internal reflection

Total internal reflection takes place when light travels from a medium with a higher refractive index to one with a lower refractive index and the angle of incidence exceeds the critical angle. The critical angle is determined by the equation n₁ sinθc = n₂, where n₁ and n₂ are the refractive indices of the denser and less dense media, respectively. When this condition is met, light is completely reflected back into the denser medium rather than refracted into the less dense one.

Spherical mirrors

Spherical mirrors are reflective surfaces derived from a sphere’s curvature and come in two forms: concave mirrors and convex mirrors:

• A concave mirror, with an inward curvature, focuses parallel light rays to a focal point, acting as a converging mirror with a positive focal length.
• A convex mirror, having an outward curvature, causes light rays to diverge and form virtual images behind the mirror, resulting in a negative focal length. The radius of curvature is the distance from the mirror to the center of the originating sphere, and the focal length is half of this value ($f=R/2$).

The imaging properties are described by the mirror equation: $1/p+1/q=1/f$, where object distance ($p$) is measured from the mirror to the object and image distance ($q$) is the distance from the mirror to the image. In these conventions, $p$ is positive, while $q$ is positive for a real image (formed in front of the mirror) and negative for a virtual image (formed behind it). Magnification ($M$) is defined as $M=h^{\prime }/h=-q/p$, with $h^{\prime }$ being the height of the image and $h$ the height of the object. Real images from concave mirrors are always inverted, whereas virtual images are upright; convex mirrors, being diverging, always produce virtual images.

Thin lenses

Thin lenses are optical devices that bend light much like mirrors, but instead of reflecting light, they transmit it to form images. There are two primary types: convex lenses (converging) and concave lenses (diverging).

A convex lens is similar to a concave mirror in that it converges light rays; real images are formed on the opposite side of the lens from the object because light passes through and focuses, while virtual images appear on the same side as the object since the light does not actually converge.

Concave lenses cause light to diverge, producing only virtual images that are upright and form on the same side of the lens as the object.

The focal length of a lens, which is half the distance of the radius of curvature for mirrors, indicates where parallel rays converge (or appear to diverge). For a converging lens, the focal length is positive, whereas for a diverging lens it is negative.

The relationship between the object distance ($P$), the image distance ($q$), and the focal length ($f$) is given by $1/p+1/q=1/f$, where $P$ is always positive, $q$ is positive for real images, and negative for virtual images.

The strength or power of a lens is measured in diopters ($P$), defined by $P=1/f$, where $f$ is measured in meters. Lenses can exhibit aberrations such as spherical aberration, where not all rays focus at a single point, and chromatic aberration, where different wavelengths (colors) refract differently, causing color fringing in images.

Combination of lenses

In an optical system, a real image produced by one lens can serve as the object for another lens. When multiple lenses are combined, the overall magnification of the system is the product of the individual magnifications from each lens.

Lens aberration

Lens aberration refers to optical system imperfections that produce blurred, distorted, or color-fringed images:

• Spherical aberration arises when rays passing through different lens regions converge at multiple points, reducing sharpness.
• Chromatic aberration occurs because the lens cannot focus all wavelengths together, creating color fringing.
• Astigmatism reflects varying focal lengths along different axes, causing uneven sharpness. Distortion means magnification changes with distance from the lens center, bending straight lines. Field curvature implies the lens projects onto a non-flat plane, blurring image edges.
• Coma leads off-axis points to appear comet-shaped, reducing clarity near the field’s periphery.

Optical Instruments

The eye functions as an optical instrument by using a lens to focus a real image onto the retina.

Glasses correct vision by employing either a diverging (concave) lens for near-sightedness or a converging (convex) lens for far-sightedness.

A magnifying glass is also a converging lens; when the object distance ($p$) is less than the focal length ($f$), it forms a virtual, erect, and enlarged image.